Scalar and vector potentials

Problem:

(a)  Write down Maxwell's equations in free space and in the presence of the current density j(r,t) and charge density ρ(r,t).  Introduce the electromagnetic potentials and derive the differential equation that they satisfy.
(b)  State the Lorentz condition and show the simplification found thereby.
(c)  Define a gauge transformation and describe its effect on the electromagnetic fields.

Solution:

Problem:

In the derivation of the wave equations for A and Φ from Maxwell's equations in a vacuum, one gets at one stage
2A - (1/c2)∂2A/∂t2 - (∇∙A +(1/c2)∂Φ/∂t ) = -μ0j.
and
2Φ + ∂∇∙A/∂t = -ρ/ε0.
This is made solvable by use of a gauge choice, using the fact that physics is invariant under a gauge transformation.  Write down the Lorentz gauge condition and the resulting PDE's for Φ and A.

Solution:

Problem:

(a)  Find the magnetic field associated with the potential A(r,t) = (b/2)n×r.
(b)  Find the charge and current distributions that would lead to
A
(r,t) = (bt/r3)r,  Φ(r,t) = 0.
(c)  Determine the charge distribution that will give rise to the potential
Φ(r,t) = W0exp(-αr)/r.

Solution:

Problem:

(a)  State Maxwell's equations and prove that they are satisfied by E = -∂A/∂t, B = ×A,  provided ∇∙A = 0, ∇2A = (1/c2)∂2A/∂t2.
(b)  Derive E and B when A = i a cos(k(z - ct))+ + j b sin(k(z - ct)).
(c)  Verify that E and B are orthogonal and their directions rotate about the z-axis with  frequency kc/(2π).

Solution:

Problem:

(a)  A point charge q rests at the origin.  
A natural choice of potentials for this static problem is V(r,t) = q/(4πε0r), A(r,t) = 0.
Consider the gauge transformation with λ(r,t) = qt/(4πε0r) + k/r3,
where k is a constant.
Calculate the transformed potentials and fields.   Discuss the result.
(b)  Find E and B given the potentials  A(ρ,φ,z) = -k C ln(ρ2)/(4π),  V = 0.
Which charge or current distributions produce these fields?   Can you find a gauge transformation which leaves the transformed vector potential  A(ρ,φ,z) = 0?

Solution:

Problem:

(a)  Write down expressions for the electric field E and the magnetic field B in terms of the scalar potential Φ and the vector potential A.
(b)  Given Φ = 0 and  A = k0α/(4c)](ct -|x|)2 for |x| < ct, A = 0 for |x| > ct, find E and B.
(c)  Write down the boundary conditions for the normal component of E and the tangential component of B at the x = 0 interface.
(d)  Find the surface charge density and the surface current density at the x = 0 interface.  (Assume media with ε0 and μ0 on both sides of the interface.)

Solution:

Problem:

Show that in free space with ρ = 0 and  j = 0 Maxwell's equations are correctly obtained from a single vector potential A satisfying

∇∙A = 0, 2A - (1/c2)∂2A/∂t2 = 0,  Φ = 0.

Solution:

Problem:

(a)  Consider the following vector potential A:
Ax = 0,  Ay = bx,  Az = 0. 
Find the corresponding magnetic field.
(b)  Consider the following vector potential A':
A'x = -by,  A'y = 0,  A'z = 0. 
Find the corresponding magnetic field.
(c)  A' = A + f(x, y,z).  Find f.
(d)  The Lagrangian of a particle with mass m and charge q moving in the presence of a magnetic field is
L = ½ mv2 + qv∙A.
Using the vector potential from part (a) or part (b), write down Lagrange's equations of motion, the Hamiltonian, and Hamilton's equations of motion.
(e)  Solve the equations of motion.